Math  /  Algebra

QuestionWhich is equivalent to 575^{-7} ? A) -35 (B) 57-5^{7}
C 5175^{\frac{1}{7}} (D) 157\frac{1}{5^{7}}

Studdy Solution

STEP 1

What is this asking? This problem is asking us to rewrite 575^{-7} with a positive exponent. Watch out! Don't be tricked by the negative exponent!
It doesn't make the number negative.

STEP 2

1. Rewrite with a Positive Exponent

STEP 3

Alright, let's tackle this negative exponent!
Remember, a negative exponent doesn't make the **base** negative.
It just flips the base to its reciprocal.
It's like a mathematical trampoline – it bounces the base to the other side of the fraction line!

STEP 4

So, we have 575^{-7}.
Our **base** is 55, and our **exponent** is 7-7.
Since the exponent is negative, we're going to flip the base.
Think of 55 as 51\frac{5}{1}.
Flipping it gives us 15\frac{1}{5}.

STEP 5

Now, we can rewrite our expression with a positive exponent: (15)7\left(\frac{1}{5}\right)^{7}.
Notice how the exponent became positive after we flipped the base!

STEP 6

We can simplify this further! (15)7\left(\frac{1}{5}\right)^{7} means we multiply 15\frac{1}{5} by itself seven times.
This is the same as 1757\frac{1^7}{5^7}.
Since 11 multiplied by itself any number of times is still 11, this simplifies to 157\frac{1}{5^7}.
Boom!

STEP 7

The equivalent expression with a positive exponent is 157\frac{1}{5^7}, so the answer is (D).

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